The double neutron star(DNS) systems originate from two supernovas resulting  from  two massive main-sequence stars under some normal conditions, basically consist of a observed radio pulsar  orbiting with a hidden neutron star(NS). The DNS is powerful laboratories to study many aspects of modern Astrophysics. Recent discoveries have confirmed the direct relations between DNS and supernova explosion. This would reflect the information about the evolutionary history of pathways of Core-Collapse or Electron-Capture Supernova for different types of DNSs. The DNS makes it valuable to allow exquisite tests of Einstein’s theory of general relativity and its prediction of the gravitational wave.
    In Chapter 1, the research progress of DNS is briefly introduced, including its discovery history, current sample size, spatial distribution, and two major formation mechanisms.
   In Chapter 2 we present the statistical  distribution of the total mass of fourteen DNS systems, and found  that it is very narrow, with the average and deviation as 2.65$\pm$0.03 M$_{\odot}$. While, for the seven pairs of DNSs with the precisely measured masses of the first-formed NSs and  second-formed companions, their  masses are homogeneously  distributed as  1.38$\pm0.02\ms$   and 1.29$\pm0.02\ms$,  respectively. Statistical tests, namely Anderson$-$Darling (A$-$D) and Mann$-$Whitney$-$Wilcoxon (M$-$W$-$W) tests, confirm the bimodality of the DNS mass distribution, which can be attributed to different formation mechanism of the first- and second-born NS in the pair. The first NS was formed through supernova explosion.
The second NS can have different types of origins, such as electron capture and ultra-stripped cores in close-orbit systems.
     In Chapter 3 we investigated the aspect of formation and evolution of the recycled  PSR J0737-3039 A, taking into account the contributions of accretion rate, radius and spin-evolution diagram  in the double pulsar system. Accepting the spin-down age as a rough estimate (or often an upper limit) of the true age of the neutron star, we also impose the restrictions on the radius of this system.  We calculate the radius of the recycled pulsar PSR J0737-3039 A ranges approximately from 8.14 to 25.74 km,
 and the composition of its neutron star nuclear matters is discussed in the mass-radius diagram. We further look into the eccentricity (e) and orbital period (P) distribution of DNSs. We obtained, through simulating two DNS populations of electron-capture and core-collapse supernova and  utilizing the Fisher’s discriminant method, a linear
discriminant relation $e  =  $-$4.5 P_{\rm orb}(day) + 1$ on the $e-P$ phase diagram.
The observed DNS sample are clearly grouped in two regions separated by this relation,
consistent with different origins of these binaries, such as electron-capture supernova (core collapse supernova)  without (with) kick.
    In Chapter 4 Based on the detected gravitational wave frequency by LIGO and the afterglow observations by the optical and Gamma-ray instruments, We present that this outcome of coalescence kilonova of GW170817 is most probably the birth of a fast rotating sub-millisecond pulsar (SMSP), with the spin frequency of $\sim $1490 Hz (period of P$\sim 0.67$ ms), and magnetic field of $B\sim 10^{11-15}$ G depending on the magnetic reconstruction process of the new NS. We  estimated the mass and radius of this new born SMSP to be about $\sim 2.70\ms$ and $12 - 25$ km, respectively,
then the EOS and nuclear composition of such  a super-spinning and massive compact object is not known, after comparing with the predictions of the known EOS models.
The spacetime around this MSP is extremely frame-dragged, since its Kerr spin parameter is expected to be as high as 0.984, almost unity like those of fast spinning black holes.The observations of this new type of MSP after GW170817 is proposed, and its extreme properties are  investigated, for instance, its magnetic dipole spin-down induced energy dissipation rate is $\dot{E}\sim 10^{44} (B/10^{12}G)^{2} erg/s,$
which is four orders of magnitudes more luminous than that of Crab pulsar, and accordingly its spin-down characteristic age is as short as 30 years. While considering the distance of the source at 130 million light years away, the corresponding energy dissipation rate is  $\dot{E}\sim 10^{34} (B/10^{12}G)^{2}$ erg/s at the location of Crab pulsar, which could be observed by sensitive radio telescopes if this pulsar sweeps its radiation beam across the earth.
     In Chapter 5, we  search and study single pulses  using the first version of database.
And we briefly describe the procedure to use Our database,
in which we can search for new FRBs or repeating events and study single pulses from known pulsars.
    we will present conclusions and outlook in Chapter 6.

    The data processing methods and meaningful candidates for single pulse involving in this paper is briefly described in appendix.